Who can give blessings? There are lay people in my parish giving blessings and this does not seem right.

Context and content are important in answering a question like this. In the liturgical setting only a priest (and sometimes the deacon) should be conferring blessings since they are present and available for such. Thus, the practice observed in some places of lay people who are distributing communion and also giving blessings is inappropriate. The priest should be sought for this, apart from the communion line. 
 
However, in other settings lay people can give certain blessings in certain ways. For example, a parent can bless a child, an elder can bless a youngster, etc. In doing this, however, they ought to avoid priestly gestures such as making the sign of the cross over others. Perhaps tracing the cross on the forehead is enough, or simply laying a hand on the head, or no gesture at all, are better. 
 
In settings where lay people are praying for one another, say in a prayer for healing or deliverance, similar rules should be followed, avoiding overt priestly gestures, and being content to lay hands, or make no gesture at all. 
 
In the rare instances where lay people lead formal liturgical gatherings, such as the Liturgy of the Hours, they must not only avoid gestures, but also follow prescribed texts which merely ask God’s blessing on the assembled believers, but do not imply they are bestowing such blessings. 
 
Finally there are certain specific prayers and blessings that can only be given by a priest. The norms are too specific to be given here in a general answer. But most blessings of objects and sacramentals are reserved to clergy, and the laity ought to be content to offer simple prayers, asking blessings for one another only in appropriate contexts.